L(s) = 1 | − 0.284·2-s + 1.42·3-s − 1.91·4-s − 5-s − 0.405·6-s − 1.39·7-s + 1.11·8-s − 0.970·9-s + 0.284·10-s + 4.82·11-s − 2.73·12-s − 0.285·13-s + 0.397·14-s − 1.42·15-s + 3.52·16-s − 0.477·17-s + 0.276·18-s − 0.989·19-s + 1.91·20-s − 1.98·21-s − 1.37·22-s + 1.58·24-s + 25-s + 0.0812·26-s − 5.65·27-s + 2.68·28-s − 0.153·29-s + ⋯ |
L(s) = 1 | − 0.201·2-s + 0.822·3-s − 0.959·4-s − 0.447·5-s − 0.165·6-s − 0.527·7-s + 0.394·8-s − 0.323·9-s + 0.0900·10-s + 1.45·11-s − 0.789·12-s − 0.0791·13-s + 0.106·14-s − 0.367·15-s + 0.880·16-s − 0.115·17-s + 0.0651·18-s − 0.226·19-s + 0.429·20-s − 0.434·21-s − 0.292·22-s + 0.324·24-s + 0.200·25-s + 0.0159·26-s − 1.08·27-s + 0.506·28-s − 0.0285·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2645 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2645 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + T \) |
| 23 | \( 1 \) |
good | 2 | \( 1 + 0.284T + 2T^{2} \) |
| 3 | \( 1 - 1.42T + 3T^{2} \) |
| 7 | \( 1 + 1.39T + 7T^{2} \) |
| 11 | \( 1 - 4.82T + 11T^{2} \) |
| 13 | \( 1 + 0.285T + 13T^{2} \) |
| 17 | \( 1 + 0.477T + 17T^{2} \) |
| 19 | \( 1 + 0.989T + 19T^{2} \) |
| 29 | \( 1 + 0.153T + 29T^{2} \) |
| 31 | \( 1 + 3.58T + 31T^{2} \) |
| 37 | \( 1 + 4.11T + 37T^{2} \) |
| 41 | \( 1 + 6.89T + 41T^{2} \) |
| 43 | \( 1 - 10.7T + 43T^{2} \) |
| 47 | \( 1 - 9.42T + 47T^{2} \) |
| 53 | \( 1 + 4.81T + 53T^{2} \) |
| 59 | \( 1 + 6.84T + 59T^{2} \) |
| 61 | \( 1 - 10.8T + 61T^{2} \) |
| 67 | \( 1 + 7.05T + 67T^{2} \) |
| 71 | \( 1 + 12.0T + 71T^{2} \) |
| 73 | \( 1 + 6.82T + 73T^{2} \) |
| 79 | \( 1 - 11.9T + 79T^{2} \) |
| 83 | \( 1 - 6.50T + 83T^{2} \) |
| 89 | \( 1 + 10.9T + 89T^{2} \) |
| 97 | \( 1 + 15.6T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.699604108758670383155930992719, −7.916159917026606704437452933595, −7.13975994604045637840448931686, −6.22356400653260906546458609536, −5.30213803088811981619886033179, −4.14842210274739764256643776175, −3.75890228027853790230578891868, −2.83461593824888427581112867501, −1.44687065269191536972675425232, 0,
1.44687065269191536972675425232, 2.83461593824888427581112867501, 3.75890228027853790230578891868, 4.14842210274739764256643776175, 5.30213803088811981619886033179, 6.22356400653260906546458609536, 7.13975994604045637840448931686, 7.916159917026606704437452933595, 8.699604108758670383155930992719